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Speed, momentum and total energy of a muon (SR)

  1. Jan 9, 2017 #1
    1. The problem statement, all variables and given/known data

    A muon has a mass of 106MeV/c2. Calculate the speed, momentum and total energy of a 200MeV muon(a muon with a kinetic energy of 200MeV).
    2. Relevant equations
    E=γmc
    K+mc2=E
    γ=1/(1-β)1/2
    β=(v/c)2

    3. The attempt at a solution
    To solve for the speed I plugged E=γmc into K+mc2=E to obtain K/mc2 + 1 = γ. Expanding gamma and rearranging I get 1-(K/mc2+1)-1 = β. But plugging in the numbers on the left hand side gives me a zero as K<<c2 and I think that I went wrong somewhere.
     
  2. jcsd
  3. Jan 9, 2017 #2

    TSny

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    The first, third, and fourth equations have errors.
    [EDIT: Sorry. Your third and fourth equations are OK if you are defining the symbol β to equal (v/c)2. But it is more common to use β for v/c.]

    E ≠ γmc. But your expression for γ looks correct.

    Check your work here, I don't think you got the correct expression for ##\beta##.

    Did you mean to write K << mc2? If so, how did you come to this conclusion based on the information given in the problem?
     
    Last edited: Jan 9, 2017
  4. Jan 10, 2017 #3
    Sorry line 1 was supposed to be E=γmc2 and sorry I skipped so many steps. I'll write down everything I did.

    K=mc2(γ-1)
    K/mc2 + 1 = 1/(√1-(v2/c2))
    1-(v2/c2)=(K/mc2 + 1)2
    1-(K/mc2 + 1)2=(v2/c2)
    c2(1-(K/mc2 + 1)2)=v2
    √[c2(1-(K/mc2 + 1)2)]=v

    did I do anything wrong?
     
  5. Jan 10, 2017 #4
    Did you mean to write K << mc2? If so, how did you come to this conclusion based on the information given in the problem?

    Well given the numbers, c2 = 9 x 1016 and from the values of K and m given K/mc2 = 0
     
  6. Jan 10, 2017 #5

    TSny

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    Check going from 2nd to 3rd line
     
  7. Jan 10, 2017 #6

    TSny

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    Note the units in the value of the mass, m =106 MeV/c2.
    So c2 cancels in mc2.
     
  8. Jan 10, 2017 #7
    I keep making mistakes as I try to type it on the site sorry and thanks.
     
  9. Jan 10, 2017 #8
    Yea I know they cancel I mean when you put in the numbers you basically get 1/∞ for E/mc2
     
  10. Jan 10, 2017 #9

    TSny

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    What numerical value do you get for mc2 (including units)?

    (Your final expression for v in post #7 looks correct.)
     
    Last edited: Jan 10, 2017
  11. Jan 11, 2017 #10
    ohh I see the c2 cancels out. Thanks!
     
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